Amida

An Amida Board B = { L, E} is defined by a sequence of N =
|L| lines (ordered left to right as L1 .. LN) and a set of
M = |E| cross-edges connecting lines Li and Li+1 at a
vertical position y. Thus a line Ej can be uniquely identified
by the triple (i, i+1, y). The vertical position is a positive integer >0 and is
referenced by the notation L.y, the two lines are L.left and L.right.

Definition: An Amida board has
unconnected lines (Li Li+1) if there is no edge Ej
in the set E such that (Ej.left = i) and (Ej.right =
i+1).

Definition: A Regular Amida
Board has no (1) unconnected lines; (2) negating cross-edges; or (3)
cross-edges with the same vertical position. The last condition (3) can easily
be assured by simply expanding the edges of the board without affecting its
outcome. Assuming this third property makes some of the definitions and
algorithms clearer.

Questions

How many potential Amida boards are there with N
lines and M cross-edges?

How many Amida boards are there with N lines and M
cross-edges?

Given any Amida board, it can be 'stretched out' so each
cross-edge Ei has a unique y value without altering the
outcome of the board. One can form a sequence of edges from the top of the board
(lowest y value) to the bottom of the board (highest y value); each of
these edges can be selected independently from the set of (N-1) potential cross
edges that can be added at that y location. Therefore, there are a total
of (N-1)W unique Amida boards. However, these Amida boards are not
guaranteed to be regular.

How many Connected Amida boards are there?

Given M cross-edges to be placed, we need to at least
place N-1 to ensure the board has no unconnected-lines. These N-1 cross-edges
can be placed (N-1)! possible ways, since the y positions of these N-1 edges
must all be different. There are (M-(N-1)) edges, or M+1-N, edges left to be
added, and these can be placed in N distinct "regions" (formed by the spaces
between the N-1 edges placed to ensure a connected Amida board.

....

How Many Regular Amida boards are there?

There are (M-(N-1)) edges, or M+1-N, edges left to be added.
The ith of these edges (0 < i < M+1-N) to be added must have a
different y position from the existing (N-2+i) edges in the board, thus
there are N-1+i possible values it could have. In addition, these edges can be
placed between different edges, but we must ensure that we do not add a negating
edge. There are a total of (N-1)*(N-1+i) potential places to place the cross
edge. Of these locations, a negating edge would occur 2*(N-2+i) times (for each
of the existing edges, a negating edge would exist just above and just below
it). Thus we can add in the following potential locations:

we now have a rough upper bound on the number of
regular Amida graphs with N lines and M cross-edges.

(M-1)! * (N-3)(M+1-N)

We must be careful not to over-count. For example, with N
edges to add, there are two distinct ways that these could be added. Thus, we
must amortize based upon the total number of cross-edges found in a single
column.